Teaching & Learning: Exploring Unsolved Problems
Teaching & Learning: Exploring Unsolved Problems
Liz Torrey

In our Teaching & Learning web series, we'll share the details of individual classroom assignments: what teachers asked their students to do and why, and how students responded to the assignment.

Advanced Topics Tutorial in Math
Taught by Jon Tower


I asked ATT Math students to select, and work on a problem from the book Old and New Unsolved Problems in Plane Geometry and Number Theory, by Victor Klee and Stan Wagon. As the name of the book implies, these are problems consist of mathematical propositions that remain unproven. Many professional mathematicians have attempted to confirm the propositions by various methods, but to date no one has been able to construct a proof of their validity. They will give presentations of their findings, including their own work on the problem at the end of this week.

The goal of this assignment is not for our students to complete a proof, a very tall order in only two weeks, but to investigate and research the problems, in much the same manner that a professional mathematician might.
—Mr. Tower

Student Responses

This was a really cool project because we were able to use the concepts we'd been learning about creating proofs this past quarter and apply them to more complex questions. For my problem, I worked on the "3n+1 conjecture" which basically says that if you take any positive integer and, if it's even, divide by 2, and if it's odd, plug it into 3n+1—either way, your answer will eventually get to 1. I think we were all challenged to use our creativity since there is no obvious way to go about the process of attempting to prove an unsolved theorem. Even though none of us were able to prove our problems, we were all able to look at the questions through a new lenses by breaking them up and simplifying them, and by the end of the two weeks, we all had a much deeper understanding of our theorems.
—Hannah Soulati '18

I worked on a problem called "the perfect cuboid", which is a box whose side lengths, face diagonals, and space diagonal are all integers. No one has ever proven that there exists such a perfect cuboid. At the first glance, this theorem looks very straightforward and simple; it involves only elements that even an elementary school student understands: a box, and whole numbers. The fact that these simple math concepts could be also be a problem that has been unsolved for centuries attracted me deeply. There's so much we as humans still don't understand and need to explore about the universe! I was able to find my own breakthroughs on the problem, and every time I figured out a small piece, I felt as if I were a part of the advancement of human knowledge. Doing this work showed me that if I really spend time on a single problem, no matter how difficult it is, I will be able to make progress.
—Leo Qiao '18